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Symmetries

September 8, 2012

We are all naturally familiar with the idea of symmetry.

Our bodies are (nearly!) symmetric. Chairs and tables, spoons and forks, pianos and fireplaces all look symmetric, even if, like a piano, they are not symmetric internally.

Symmetry is an also idea which is fundamental to mathematics. Some simple examples are:

  • The end digits of the square numbers (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400…) follow a cycle of 0,1,4,9,6,5,6,9,4,1,0,1,4,9,6,5,6,9,4,1,0,…
  • Isosceles triangles (like Δ) have two equal sides and two equal angles.
    Many ideas in trigonometry come out of working with isosceles triangles.
  • Theories of sub-atomic physics are critically dependent on symmetry principles.
    For example there are particles (e.g. electrons) and anti-particles (e.g. positrons).
    These are identical, except that certain parameters have opposite values.

1. Symmetries of the alphabet

Let’s take a simple example of symmetries of upper case  (capital) letters.

Those with a horizontal line of symmetry are BCDEHIKO.
Those with a vertical line of symmetry are AHIMOTUVWXY.
Those with (180°) rotational symmetry are HINOSZ.

If we arrange these in a table, as below, then count the symmetries of each letter, we find a surprising lack of any letter with 2 (but not 3) symmetries. That is, if a letter has two of the three symmetries, then it must have the other symmetry as well. In fact, this applies not just to letters but to any two dimensional shape. Try it out with other shapes.

So this connection is a property of the three symmetries themselves, not of the letters (or other shapes) that possess them.

Letter

Symmetries

Horizontal

Vertical

Rotational

Count

A

Yes

1

B

Yes

1

C

Yes

1

D

Yes

1

E

Yes

1

F

0

G

0

H

Yes

Yes

Yes

3

I

Yes

Yes

Yes

3

J

0

K

Yes

1

L

0

M

Yes

1

N

Yes

1

O

Yes

Yes

Yes

3

P

0

Q

0

R

0

S

Yes

1

T

Yes

1

U

Yes

1

V

Yes

1

W

Yes

1

X

Yes

1

Y

Yes

1

Z

Yes

1

2. Calendar Puzzle

As another example of symmetry, this is stretching it a little, but is something similar for numbers and lower case letters…

We can represent the date as ‘ddmmmyy’, with ‘dd’ as the day of the month, ‘mmm’ as the the first three lower case characters of the month of the year (you can’t do it with upper case!), and ‘yy’ as the last two digits of the year. An example is ’08sep12′.

So, make a cube calendar, as follows:

  • Put numbers on the faces of two cubes to make every day of the month from ’01’ to ’31’.
  • Put letters on the faces of three cubes to make the first three letters of every month, from ‘jan’ to ‘dec’. (Hint: You’ll need to use the transformation of some letters to others.)
  • Put numbers on the faces of two cubes to make every year from ’12’ (2012) to … hmm, how far can you make your calendar last? I reckon I’ll be 84 before I’d need to make another one.

I’l give my take on this next week. Put it in your calendar!

Note: The ‘months’ cubes are possible in many (but not all) languages.

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One Comment
  1. I love your posts Mark,

    PersonalIy I like the symmetry of the end digits of the times tables:
    0 times table (trivial but bear with me) has the set of end digits of ‘0’.
    1 times table (trivial again, please bear with me) has the repeating set of end digits (0,1,2,3,4,5,6,7,8,9),
    2 times table has the repeating set of (0,2,4,6,8)
    3 times table has the repeating set of (0,3,6,9,2,5,8,1,4,7)
    4 times table has the repeating set of (0,4,8,2,6)
    5 times table has the repeating set of (0,5)
    6 times table has the repeating set of (6,2,8,4,0)
    7 times table has the repeating set of (7,4,1,8,5,2,9,6,3,0)
    8 times table has the repeating set of (8,6,4,2,0)
    9 times table has the repeating set of (9,8,7,6,5,4,3,2,1,0)

    Of course, 10 repeats 0, 11 repeats 1, 12 repeats 2, etcetera. Notice now that 0 is not trivial, it is part of the same repeating pattern of end digits.

    But there is more:
    – we have 1 repeating set of 1 end digit (0) to start off our decimal set of repeating numbers.
    – we have 1 repeating set of 2 unique ending digits (0,5) 4 numbers after the repeat 0 digit.
    – we have 4 repeating sets of 5 unique ending digits (0,2,4,6,8)
    – we have 4 repeating sets of 10 unique ending digits (0,1,2,3,4,5,6,7,8,9)

    … and there is another symmetry:
    taking either the 0 or the 5 times table as our mirror we see mirror images of the 5 unique ending digits and of the 10 unique ending digits:
    – the 4 and 6 times table ending digit patterns are mirror images of each other.
    – the 2 and 8 times table ending digit patterns are mirror images of each other.
    – the 7 and 3 times table ending digit patterns are mirror images of each other.
    – the 1 and 9 times table ending digit patterns are mirror images of each other.
    … and notice that they are equal distances from the 0 or 5 times table ending pattern.

    I find all of this beautiful.

    Jeremy

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